Geometrically exact beam finite element formulated on the special Euclidean group

Sonneville, Valentin; Cardona, Alberto; Brüls, Olivier

doi:10.1016/j.cma.2013.10.008

Cited by 146 publications

(172 citation statements)

References 32 publications

(62 reference statements)

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“…The relevance of SE(3) and the associated algebra se(3) to rod theory is acknowledged by Sander [18], discussed in a review by Chirikjian [19], and exploited in beam modeling by Sonneville, Cardona and Brüls [20,21]. These authors base their approaches on representing a rod through elements of the group SE(3), but we propose that the algebra se(3) provides a representation that is more naturally and directly related to the shape of a rod.…”

Section: The Lie Algebra and The Lie Group Associated With The Rod Dementioning

confidence: 99%

“…3, we introduce a discretization of the shape of a rod viewed as a path in the algebra se(3), as defined by L. This generalizes to special Cosserat rods the approach used by Bertails, Audoly, Cani, Querleux, Leroy and Lévêque [11] for Kirchhoff rods and can be viewed as a bridge between the methods of Sander [18], Chirikjian [19], and Sonneville, Cardona and Brüls [20,21], based on the special Euclidean group, and the aforementioned ones, based on Lie algebraic quantities. A distinguishing feature of the present approach is that it obviates the need for any interpolation associated with the reconstruction of the shape of a rod from a finite sampling of its placement in space.…”

Section: The Lie Algebra and The Lie Group Associated With The Rod Dementioning

confidence: 99%

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Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

Giusteri

Fried

2017

J Elast

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We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves.

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Section: The Lie Algebra and The Lie Group Associated With The Rod Dementioning

confidence: 99%

Section: The Lie Algebra and The Lie Group Associated With The Rod Dementioning

confidence: 99%

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

Giusteri

Fried

2017

J Elast

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“…In this context the spatial description (referred to as fixed-pole formulation) has proven to be beneficial [31]. Recent results on Lie group modeling of beams can be found in [77,78].…”

Section: Geometric Integrationmentioning

confidence: 99%

Screw and Lie group theory in multibody dynamics

Müller

2017

Multibody Syst Dyn

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Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS), and at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows to use arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper, recursive O(n) Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. Two forms of MBS motion equations are derived in closed form using the Lie group formulation: the so-called Euler-Jourdain or "projection" equations, of which Kane's equations are a special case, and the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

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“…Due to the relatively low computational complexity of a single element, this restriction can often be compensated for by a finer discretization of the beam model. For example, Sonneville, Cardona, and Brüls [28] achieve good convergence characteristics and small approximation errors for a standard test case based on a bent Cantilever beam subjected to a fixed load.…”

Section: Interplay Of Rotations and Translationsmentioning

confidence: 99%

“…the Lie algebra se(3) associated with the special Euclidean group SE(3) := SO(3) Id T (3, R), the semidirect product of rotations and translations in three dimensions [28]. By identifying the local material frames, i.e.…”

Section: Interplay Of Rotations and Translationsmentioning

confidence: 99%

Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

Schröppel

Wackerfuß

2016

Adv. Model. and Simul. in Eng. Sci.

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We propose a novel finite element formulation that significantly reduces the number of degrees of freedom necessary to obtain reasonably accurate approximations of the low-frequency component of the deformation in boundary-value problems. In contrast to the standard Ritz-Galerkin approach, the shape functions are defined on a Lie algebra-the logarithmic space-of the deformation function. We construct a deformation function based on an interpolation of transformations at the nodes of the finite element. In the case of the geometrically exact planar Bernoulli beam element presented in this work, these transformation functions at the nodes are given as rotations. However, due to an intrinsic coupling between rotational and translational components of the deformation function, the formulation provides for a good approximation of the deflection of the beam, as well as of the resultant forces and moments. As both the translational and the rotational components of the deformation function are defined on the logarithmic space, we propose to refer to the novel approach as the "Logarithmic finite element method", or "LogFE" method.

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Geometrically exact beam finite element formulated on the special Euclidean group

Cited by 146 publications

References 32 publications

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

Screw and Lie group theory in multibody dynamics

Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

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